Homework 3. Cross Country Estimation of Health Production in Theory and
Practice. . Due: Thursday, October 8th.
From the homework 1:
Consider two health outcomes with two associated production functions. The first is the health of the
very young as measured by the infant mortality rate (denote the infant mortality rate as π). The second
outcome is the health of adult women as measured by the 5 year survival rate for breast cancer (denoted
as π ). In addition to these two outcomes, we are going to assume there are only two inputs into health.
This first input is labor (as measured by physicians per 1000 population), denoted by π. The second input
is capital measured by health expenditures as a % of GDP1 and denoted by π.
Given this set up, assume the following production functions:
β’ Infant Health Production: π = π΄π!π”. where A is a constant multiplier on the production
function and πΌ and π½ are the Cobb Douglas exponents which denote the relative efficiency of
labor and capital for the production of infant health.
β’ Adult Female Health Production: π = π΅π#π$. where B is a constant multiplier on the production
function and πΎ and πΏ are the Cobb Douglas exponents which denote the relative efficiency of
capital and labor.
β’ Assume that the price of capital is some constant π€%, the price of labor is some constant π€&
(which we could assign as the marginal cost of training and hiring an additional physician per
1000), and that society is willing to pay some market price to decrease the infant mortality rate by
1 per 1000 births, and that price is denoted as π`(. Likewise society is willing to pay some
market price to increase the breast cancer survival rate by 1% (denoted as π)).
1. Finally, we are ready to estimate a national cobb-Douglas health production function. Here I have
estimatd the theoretical model of infant health that we developed in homework 1.
Lnphys1000_imp: is log physicians per 1000 people in the country. Lnhlthgdp: is log of health
expenditures as a % of GDP. Otherwise, use the description of the production functions from
problem 3 of homework 1 (reprinted above) to answer these questions.
1 Holding physicians per capita constant.
_cons 2.777886 .810109 3.43 0.002 1.125658 4.430115
lnhlthgdp -.2912145 .3751454 -0.78 0.443 -1.056329 .4738996
lnphys1000_imp -.6822464 .3636548 -1.88 0.070 -1.423925 .0594325
lninfmort Coef. Std. Err. t P>|t| [95% Conf. Interval]
Total 6.53903266 33 .198152505 Root MSE = .42227
Adj R-squared = 0.1001
Residual 5.52775327 31 .178314622 R-squared = 0.1547
Model 1.01127939 2 .505639695 Prob > F = 0.0740
F(2, 31) = 2.84
Source SS df MS Number of obs = 34
. reg lninfmort lnphys1000_imp lnhlthgdp
a. What is the policy impact on infant mortality of training and hiring additional doctors such
that the physicians per 1000 heads within a country increases by 1%?
Now I estimate the parameters on the production function for adult female health.
Lnbcancer_imp is the log of the 5 year breast cancer survival rate, otherwise use the above
descriptions:
b. Suppose we want to know a lower bound estimate of the percent increase in the breast
cancer survival rate caused by a 1% increase in medical expenditures (as a percent of GDP).
According to the 95% Confidence Interval what is this estimate? Does the confidence
interval include zero? What is the significance of whether the confidence interval includes
(or excludes) zero?
.
_cons -.6392901 .1007106 -6.35 0.000 -.8446908 -.4338894
lnhlthgdp .1856352 .0466371 3.98 0.000 .0905182 .2807522
lnphys1000_imp .0359416 .0452086 0.80 0.433 -.056262 .1281452
lnbcancer_imp Coef. Std. Err. t P>|t| [95% Conf. Interval]
Total .142566123 33 .004320186 Root MSE = .0525
Adj R-squared = 0.3621
Residual .085430396 31 .002755819 R-squared = 0.4008
Model .057135727 2 .028567863 Prob > F = 0.0004
F(2, 31) = 10.37
Source SS df MS Number of obs = 34
. reg lnbcancer lnphys1000_imp lnhlthgdp
c. Assume expenditures as a percent of GDP is fixed at 6. Assume that the market price of
increasing the breast cancer survival probability by 1 percent is $100,000,000 and the cost of
training and hiring an additional physician per 10000 people is $2,000,000. According to
your derivations of the first order conditions in problem 3 of homework 1, what is the
efficient level of labor? What is the efficient 5 year breast cancer survival rate?
d. Now assume that we have 10 physicians per 1000 people within the US. These physicians
can only either be allocated to servicing infants (and therefore are inputs into the infant
health production) or breast cancer patients (and therefore are inputs into the adult female
health production function). Assume that in both production functions health expenditures
as a percent of GDP is 6. Assume that there always must be a minimium of 1 physcian per
1000 in each of the production functions. Graph the production possibility frontier showing
the trade off between infant health and adult female health. It is just fine to plot 5 or 6
points on the frontier and then trace out the general shape. Note that this PPF looks a little
different than what we typically see in an econ text book, why?
e. Choose any point on the PPF as your preferred point for allocative efficiency. Just tell me
what that means society is producing and how this point would reflect societiesβ values.
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